A conditioned local limit theorem for non-negative random matrices

Let $(S_n{)}_n$ be the random process on $\mathbb{R}$ driven by the product of i.i.d. non-negative random matrices and $\tau$ its exit time from $]0,+\mathrm{\infty }[$. By using the adapted strategy initiated by D. Denisov and V. Wachtel, we obtain an asymptotic estimate and bounds of the probability that the process $(S_k{)}_k$ remains non negative up to time $n$ and simultaneously belongs to some compact set $[b,b+\mathrm{\ell }]\subset {\mathbb{R}}^{\ast +}$ at time $n$.